Quantaloids, enriched categories and automata theory
type: Item
instance of: scholarly article
title: Quantaloids, enriched categories and automata theory
zbMATH Open document ID: 0833.18002
DOI: 10.1007/BF00878445
published in: Applied Categorical Structures
publication date: 1996-02-26T00:00:00Z
review text: Locales have been suggested as substitutes for topological spaces. Quantales provide one approach to ``non-commutative spaces''; but, as usual in non-commutative situations, it is natural to consider a ``several objects'' version (as \textit{B. Mitchell} has considered additive categories, called ``ringoids'', instead of non-commutative rings). Quantaloids are themselves categories enriched in the monoidal category of sup-lattices, and yet they also can be used as base bicategories for enriching over. This survey paper avails itself of enriched category theory at both these levels to study its subject and its application to tree automata and context-free languages. It was Betti's work [cf. \textit{R. Betti}, ``Automi e categorie chiuse'', Boll. Unione Mat. Ital., V. Ser., B 17, 44-58 (1980; Zbl 0456.18003)] on automata, boosted by Walters' identification of sheaves as enriched categories [cf. \textit{R. F. C. Walters}, ``Sheaves on sites as Cauchy-complete categories'', J. Pure Appl. Algebra 24, 95-102 (1982; Zbl 0497.18016)] which attracted the development of enrichment over a base bicategory. It is interesting to see in this paper how that general develoment can be fed back to gain further insight on automata by guiding the formulation of precise definitions and concepts.
Mathematics Subject Classification ID: 18B20
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 68Q70
Mathematics Subject Classification ID: 18B35
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 807879
zbMATH Keywords: quantaloids
zbMATH Keywords: enriched categories
zbMATH Keywords: bicategories
zbMATH Keywords: context-free languages
zbMATH Keywords: automata
author: K. I. Rosenthal
reviewed by: Ross H. Street
MaRDI profile type: Publication
cites work: Quantales, observational logic and process semantics
cites work: Q3904750
cites work: Q3658172
cites work: Q3961001
cites work: Q4721656
cites work: Variation through enrichment
cites work: Order ideals in categories
cites work: An axiomatics for bicategories of modules
cites work: Q4132170
cites work: Modal logics withn-ary connectives
cites work: Q3965241
cites work: Q3667049
cites work: Q3028349
cites work: Q3978743
cites work: Q3934579
cites work: FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES
cites work: Metric spaces, generalized logic, and closed categories
cites work: Constructing locales from quantales
cites work: Q3769981
cites work: Applications of Sup-Lattice Enriched Category Theory to Sheaf Theory
cites work: Q3996704
cites work: Free quantaloids
cites work: Girard quantaloids
cites work: Quantaloidal nuclei, the syntactic congruence and tree automata
cites work: Q4285618
cites work: A categorical look at tree automata and context-free languages
cites work: \(*\)-autonomous categories of bimodules
cites work: ENRICHED CATEGORIES AND COHOMOLOGY
cites work: Cauchy characterization of enriched categories
cites work: Sheaves on sites as Cauchy-complete categories
cites work: A note on context-free languages
cites work: The free category with products on a multigraph
cites work: Categories and Computer Science
description: scientific article; zbMATH DE number 807879